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On the finiteness of $\mathfrak{P}$-adic continued fractions for number fields

Laura Capuano, Nadir Murru, Lea Terracini

TL;DR

The paper builds a unified framework for $\mathfrak{P}$-adic continued fractions over number fields by introducing types $\tau=(K,\mathfrak{P},s)$ that generalize Browkin and Ruban. It proves finiteness results (CFF) for norm-Euclidean fields with Euclidean minimum $M(K)<1$, showing almost all prime ideals yield finite expansions and linking finiteness to the ideal class group structure. It provides explicit constructions in quadratic fields, including $\mathbb{Q}(\sqrt{2})$ and imaginary quadratic fields, and clarifies how CFF interacts with Euclidean ideal classes. The work also outlines open questions on effectiveness, explicit type construction, and periodicity, framing future directions in non-archimedean continued fractions.

Abstract

For a prime ideal $\mathfrak{P}$ of the ring of integers of a number field $K$, we give a general definition of $\mathfrak{P}$-adic continued fraction, which also includes classical definitions of continued fractions in the field of $p$--adic numbers. We give some necessary and sufficient conditions on $K$ ensuring that every $α\in K$ admits a finite $\mathfrak{P}$-adic continued fraction expansion for all but finitely many $\mathfrak{P}$, addressing a similar problem posed by Rosen in the archimedean setting.

On the finiteness of $\mathfrak{P}$-adic continued fractions for number fields

TL;DR

The paper builds a unified framework for -adic continued fractions over number fields by introducing types that generalize Browkin and Ruban. It proves finiteness results (CFF) for norm-Euclidean fields with Euclidean minimum , showing almost all prime ideals yield finite expansions and linking finiteness to the ideal class group structure. It provides explicit constructions in quadratic fields, including and imaginary quadratic fields, and clarifies how CFF interacts with Euclidean ideal classes. The work also outlines open questions on effectiveness, explicit type construction, and periodicity, framing future directions in non-archimedean continued fractions.

Abstract

For a prime ideal of the ring of integers of a number field , we give a general definition of -adic continued fraction, which also includes classical definitions of continued fractions in the field of --adic numbers. We give some necessary and sufficient conditions on ensuring that every admits a finite -adic continued fraction expansion for all but finitely many , addressing a similar problem posed by Rosen in the archimedean setting.

Paper Structure

This paper contains 20 sections, 21 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Notations and prerequisites
  3. $\mathfrak{P}$-adic continued fractions
  4. $\mathfrak{P}$-adic floor functions and types
  5. Types arising from generators of $\mathfrak{P}$
  6. $\mathfrak{P}$-adic continued fractions associated to types
  7. The quality of the $\mathfrak{P}$-adic approximation
  8. Finiteness and periodicity
  9. Browkin expansion of rational numbers
  10. Ruban expansion of rational numbers
  11. The main tools
  12. A particular case: the $\mathop{\mathrm{CFF}}\nolimits$ and $\mathop{\mathrm{CFP}}\nolimits$ properties for special types
  13. The $\mathop{\mathrm{CFF}}\nolimits$ property for norm Euclidean number fields
  14. Euclidean minimum
  15. The $\mathop{\mathrm{CFF}}\nolimits$ property for norm Euclidean number fields
  16. ...and 5 more sections

Key Result

Proposition 2.2

For every non-zero $x, y \in \overline{\mathbb{Q}}$, we have:

Theorems & Definitions (49)

  • Definition 2.1
  • Proposition 2.2
  • Definition 3.1
  • Theorem 3.2
  • Example 3.3: Browkin and Ruban types over $\mathbb{Q}$.
  • Remark 3.4
  • Definition 3.5
  • Proposition 3.6
  • Proposition 3.7
  • proof
  • ...and 39 more